Thin stationary sets and disjoint club sequences

Friedman, Sy-David; Krueger, John

doi:10.1090/S0002-9947-06-04163-8

Thin stationary sets and disjoint club sequences
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by Sy-David Friedman and John Krueger PDF

Trans. Amer. Math. Soc. 359 (2007), 2407-2420 Request permission

Abstract:

We describe two opposing combinatorial properties related to add- ing clubs to $\omega _2$: the existence of a thin stationary subset of $P_{\omega _1}(\omega _2)$ and the existence of a disjoint club sequence on $\omega _2$. A special Aronszajn tree on $\omega _2$ implies there exists a thin stationary set. If there exists a disjoint club sequence, then there is no thin stationary set, and moreover there is a fat stationary subset of $\omega _2$ which cannot acquire a club subset by any forcing poset which preserves $\omega _1$ and $\omega _2$. We prove that the existence of a disjoint club sequence follows from Martin’s Maximum and is equiconsistent with a Mahlo cardinal.

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